A216702 - OEIS

%I #37 Apr 02 2024 11:44:27

%S 1,12,168,2464,36960,561792,8614144,132903936,2060011008,32044615680,

%T 499896004608,7816555708416,122459372765184,1921670157238272,

%U 30197673899458560,475110069351481344,7482983592285831168,117967035454858985472,1861257670509997326336

%N a(n) = Product_{k=1..n} (16 - 4/k).

%C This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

%H Harvey P. Dale, <a href="/A216702/b216702.txt">Table of n, a(n) for n = 0..800</a>

%F G.f.: 1/(1-16*x)^(3/4). - _Harvey P. Dale_, Sep 19 2012

%F From _Peter Bala_, Sep 24 2023: (Start)

%F a(n) = 16^n * binomial(n - 1/4, n).

%F P-recursive: a(n) = 4*(4*n - 1)/n * a(n-1) with a(0) = 1. (End)

%F From _Peter Bala_, Mar 31 2024: (Start)

%F a(n) = (-16)^n * binomial(-3/4, n).

%F a(n) ~ 1/Gamma(3/4) * 16^n/n^(1/4).

%F E.g.f.: hypergeom([3/4], [1], 16*x).

%F a(n) = (16^n)*Sum_{k = 0..2*n} (-1)^k*binomial(-3/4, k)* binomial(-3/4, 2*n - k).

%F (16^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).

%F Sum_{k = 0..n} a(k)*a(n-k) = (16^n)/(2*n)! * Product_{k = 1..n} (4*k^2 - 1) =  (16^n)/(2*n)! * A079484(n). (End)

%p seq(product(16-4/k, k=1.. n), n=0..20);

%p seq((4^n/n!)*product(4*k+3, k=0.. n-1), n=0..20);

%t Table[Product[16-4/k,{k,n}],{n,0,20}] (* or *) CoefficientList[ Series[ 1/(1-16*x)^(3/4),{x,0,20}],x] (* _Harvey P. Dale_, Sep 19 2012 *)

%Y Cf. A004988, A049382, A004994, A034385, A098430.

%K nonn,easy

%O 0,2

%A _Michel Lagneau_, Sep 16 2012